Outline of a Theory of Functions of an Abstract Variable 79 
form curve contained in R. Then 
J sear = 0. 
Proof. Since f(x) isan extended regular function (see T.38), 
it follows that the curve may be separated into a finite number 
of arcs on each of which the function is represented by a single 
power series. If there are 7 such arcs we may write — 
f feax =f flaydx + f : f(a)dx +--+ +f f(x)dx 
r Fe, Te; T'za—; 
where xo, Xi, - - + , X,-1 mark the end points of the arcs, and ap- 
plying the result of T.34 to each integral, we find that the terms 
cancel each other in succession. 
In particular, of course, the curve I’ may lie entirely within 
a fixed complex-plane and in this case we have also the formulas 
proved by Gateaux (assuming f(x) to be regular at x~- 0 for 
convenierce): 
1 f(éx)dé 
fls) = 5 cones 
which holds for all xeR if I’ encloses the origin; and in the same 
region 
Qirtn}x” = 
f(&x) : 
Jrivr, °° 
Gateaux has also proved the following theorem 
7.41. If 
= Do atest I| xl] <p 
and u(p:) is the least upper bound of || ae sethe sphere || || 
=pi, then 
u(p1) 
pr 
|| @tr4n1 2" = 
for all values of n and all x. 
From this last we may deduce a theorem analogous to that 
of Liouville, namel 
1.41. A faaee which is regular and bounded throughout 
